Dice with curved, but non-spherical surfaces, certain of which feature nonuniform display probabilities

ABSTRACT

The current document is directed to gaming dice, including physical dice and virtual dice represented on an electronic display or viewing screen. Certain of the currently disclosed gaming dice have approximately ovoid, prolate spheroid, or ellipsoid shapes that are modified to include planar surfaces. Other of the disclosed gaming dice have angular shapes, including cubes, tetrahedrons, octahedrons, and other regular shapes that are modified to provide nonuniform probabilities of position landings.

CROSS-REFERENCE TO RELATED APPLICATION

This application claims the benefit of Provisional Application No. 62/388,414, filed Jan. 30, 2016.

COPYRIGHT NOTICE AND PERMISSION

This document contains some material which is subject to copyright protection. The copyright owner has no objection to the reproduction with proper attribution of authorship and ownership and without alteration by anyone of this material as it appears in the files or records of the Patent and Trademark Office, but otherwise reserves all rights whatsoever.

TECHNICAL FIELD

The present invention relates to the fields of entertainment and gaming and, in particular, to multifaceted dice that, after thrown or rolled, land on at least two different planar surfaces with two different probabilities.

BACKGROUND

Multifaceted dice with indices on at least one surface have been used for games and gambling for thousands of years. The shape most familiar is a cube with the faces having indices on the surfaces such as numbering with the numerals 1 through 6 or with dots representing those numbers. The dice faces have also been decorated with special marks or indicia which distinguish the different surfaces of the die, including images, colors, and various icons, shapes and symbols. In more recent years, non-cubic dice with numbered or specially marked faces have found favor with people interested in role playing games where expanded probabilities are desirable. Platonic solids having 4 to 20 regular faces of triangles, squares and pentagons are commonly used in these games. Such types of dice typically have planar faces of equal size and shape and the surfaces are generally provided with marks or indicia. Non-polyhedral shapes have been used also. For example, a spherical die has been developed. The spherical die has the numbers 1-6, or other markings, located equidistant from one another on spherical surface. To make the die land stably, the die is provided with a hollow interior formed in the shape of a regular octahedron, which has 6 equidistant vertices, and an internal metal ball bearing to assure that one of the rounded surfaces with a mark or number will face upward when the die comes to rest.

While most die are designed with sharp edges, it is possible to round the corners and edges, either by a manufacturing process or by tumbling the dice in an abrasive media following manufacture. Examples of the above-mentioned types of dice are illustrated in FIG. 1. While not known to be normally used as gaming dice, Archimedean solids, which are generally derived from Platonic solids but have different shapes on their surfaces, such as, for example, more than one type of regular triangle, square or polygon surface, can also be used for gaming and entertainment. However, because of their greater complexity and their lower predictability in terms of results, solid Archimedean polygons are unlikely to be useful in most gaming applications, but may find use in gaming applications where nonuniform probabilities among the possible landing or resting positions of an Archimedean-polygon die are acceptable or desirable. Examples of solid Archimedean polygons are shown in FIG. 2

BRIEF SUMMARY

The current document is directed to gaming dice, including physical dice and virtual dice represented on an electronic display or viewing screen. Certain of the currently disclosed gaming dice have approximately ovoid, prolate spheroid, or ellipsoid shapes that are modified to include planar surfaces. Other of the disclosed gaming dice have angular shapes, including cubes, tetrahedrons, octahedrons, and other regular shapes that are modified to provide nonuniform probabilities of position landings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIGS. 1A-G provide examples of currently available gaming dice.

FIG. 2A-M provides examples of Archimedean-polygon dice.

FIGS. 3A-C show a die with 3 primary and equally probable landing surfaces.

FIGS. 4A-C show a die with 4 primary and equally probable landing surfaces.

FIGS. 5A-C show a die with 5 primary and equally probable landing surfaces.

FIGS. 6A-E show a die with 3 primary and equally probable landing surfaces and 2 additional surfaces having at least one lower probability.

FIGS. 7A-E show a die with 4 primary and equally probable landing surfaces and 2 additional surfaces having at least one lower probability.

FIGS. 8A-E shows a die with 5 primary and equally probable landing surfaces and 2 additional surfaces having at least one lower probability.

FIGS. 9A-B shows a polyhedron die with 4 primary and equally probable landing surfaces and 4 additional surfaces having at least one lower probability.

DETAILED DESCRIPTION

FIGS. 1A-G provide examples of currently available gaming dice and shapes of dice. FIG. 1A shows a tetrahedral die shape having four sides. FIG. 1B shows a hexahedral, or cubic, die shape having six sides. FIG. 1C shows an octahedral die shape having eight sides. FIG. 1D shows a dodecahedral die shape having 12 sides. FIG. 1E shows an icosahedral die shape having 20 sides. The shapes shown in FIGS. 1A-E are referred to as the “Platonic solids.” FIG. 1F shows an example of a round die having six numbered locations on the surface, the six locations corresponding to the 6 vertices of an octahedral cavity in the interior of the sphere. A weight, such as a ball bearing, provides a mechanism for holding the die in a stable position when the die comes to rest after being rolled or thrown. FIG. 1G shows another example of a die with rounded edges and corners.

FIG. 2A-M provides examples of Archimedean-polygon dice. As discussed above, these types of dice may be used in applications in which is it desirable for two or more landing positions of the die associated with different probabilities are desired.

In this discussion, a landing position is the position of a die when it has come to rest, after being thrown or rolled. A die in a landing position is generally resting on a planar portion of the die's surface, referred to as the “landing surface,” with one marking or indicia on a complementary, generally opposite surface presented as the highest marking or indicia that represents a result of outcome of the throw or roll. In standard, hexahedral dice, the result or outcome is printed or inscribed on the top, horizontal surface opposite from the landing surface. However, in certain of the dice to which the current document is directed, markings or other indicia that represent the result or outcome may be printed on, or inscribed in, a curved surface or on a planar surface that is not horizontally oriented in the resting position of the dice. In general, however, the markings or indicia representing the result or outcome are located in a most prominent position when the dice are in resting position, so that the result or outcome is unambiguous.

There is a probability associated with each result or outcome. For a well-manufactured standard cubic die, the probability that any particular number of the numbers 1-6 results from a roll is 1/6. The probability of a particular result is the probability of the die coming to rest in a stable landing position on the opposite, complementary face or planar surface, so that the markings or indicia representing the result or outcome are visible in a most prominent location, generally the highest position with respect to the surface on which the die rests. For a Platonic solid with n faces, in which the faces have identical shapes and sizes, and in which the same number of faces include each vertex point, the probability that die lands on a particular face after being thrown or rolled is referred to as “the probability associated with the face” or “the probability of the face,” which is also the probability that the result of the throw or roll is the marking or indicia on the, opposite, complementary side. Similarly, in dice, discussed below, the probability that a die lands on a particular planar surface is referred to as “the probability associated with the surface” or “the probability of the surface,” which is also the probability that the result of the throw or roll is the marking or indicia in the opposite, complementary position, the most visually prominent position on the resting die.

In certain of the implementations, discussed below, a die has a die body with a non-spherical, curved surface and two or more planar landing surfaces. In these implementations, a result indication is associated with each planar landing surface, each result indication prominently displayed when the die rests in a stable position on the planar landing surface with which the result indication is associated. In certain of the implementations, discussed below, the non-spherical, curved surface of the die body comprises a portion of the surface of one of an ovoid-shaped solid, a prolate-spheroid-shaped solid, and an ellipsoid-shaped solid. In certain of the implementations, the result indication associated with a planar landing surface is located at a position at the intersection of the surface of the die body and a line normal to the planar landing surface that passes through the center of the planar landing surface and continues through the die body to the position of the result indication. In certain of the implementations, discussed below, each result indication comprises a numeral, marking, or indicia that is inscribed in, molded onto, printed on, painted on, and/or affixed to the surface of the die in a neighborhood surface about the result-indication position. In certain of the implementations, discussed below, the landing surfaces are symmetrically disposed about a rotation symmetry axis of a solid, with which the curved surface of the die body is coincident. In certain of the implementations, discussed below, the landing surfaces have a common shape and size and are associated with equal landing-surface probabilities. In certain of the implementations, discussed below, a first set of two or more of the landing surfaces are symmetrically disposed about a rotation symmetry axis of a solid, with which the curved surface of the die body is coincident and a second set of two of the landing surfaces are normal to the rotation symmetry axis of the a solid, with which the curved surface of the die body is coincident. In certain of the implementations, discussed below, the landing surfaces include a first set of two or more landing surfaces that have a common first shape and first size and are associated with equal first landing-surface probabilities and a second set of two or more landing surfaces that have a common second shape and second size and are associated with equal second landing-surface probabilities that differ from the first landing-surface probabilities.

FIGS. 3A-C show a die with 3 primary and equally probable landing surfaces. FIG. 3A provides a side view of an ellipsoid-based gaming die 300 with three identically sized and shaped planar surfaces 301 designed to provide three different but equally possible outcomes indicated by a numeral, markings, or other indicia 302 on the curved surfaces opposite the planar surfaces, such as the curved surface 303 opposite the planar surface 301′. The elongation of the die along the axis of symmetry of the ellipsoid and the resulting slightly pointed ends helps better ensure that die comes to rest on one of the planar surfaces, just as it is very difficult to balance an egg or football on end without support.

FIG. 3B provides a perspective view of the die 300 in a landed position. FIG. 3C provides an end view of the die with one of the three planar surfaces 301′ in a landed position.

FIGS. 4A-C show a die with 4 primary and equally probable landing surfaces. FIG. 4A provides a side view of an ellipsoid-based gaming die 400 with four identically shaped and sized planar surfaces 401 and 401′ designed to provide four different but equally possible outcomes. Each planar surface includes numbers, markings, or indicia 402. The elongation of the die and the resulting slightly pointed ends, such as end 403, helps to better ensure that die comes to rest on one of the four planar surfaces. As stated earlier, it is very difficult to balance an egg or football on end without support.

FIG. 4B provides a perspective view of the die 400 in a landed position and FIG. 4C provides and end view of the gaming die 400 with one of the four planar surfaces 401′ in a landed position. The four planar surfaces are visible, on end. One of the most curved, or pointed, surfaces is pointing upward, out of the plane of the page. Ellipsoid surfaces 404 connect the planar surfaces where, on a cubic die, the planar surfaces would meet at angles of 90°.

FIGS. 5A-C show a die with 5 primary and equally probable landing surfaces. FIG. 5A provides a side view of an ellipsoid-based gaming die 500 with five identically shaped and sized planar surfaces 501 designed to provide five different but equally possible outcomes. An outcome is indicated by the number, marking, or other indicia 502 on the curved side 503 of the ellipsoid and opposite the planar surface 501′ shown in a landing position. The elongation of the die and the resulting slightly pointed end helps to better ensure that die comes to rest on one of the five planar surfaces.

FIG. 5B provides a perspective view of the die 500 in a landed position. FIG. 5C provides an end view of the ellipsoid gaming die 500.

FIGS. 6A-E show a die with 3 primary and equally probable landing surfaces and 2 additional surfaces having at least one lower probability. FIG. 6A provides a top view of an ellipsoid-based gaming die 600 having planar surfaces 602 with a roll-result indicium 603 facing upward. In FIG. 6A, the indicium comprises two dots representing the number 2. FIG. 6A also shows that the ends of die 600 have been truncated to generate two additional planar surfaces, including planar surface 604, at the pointed ends of the ellipsoid. These additional planar surfaces are also landing surfaces, but the additional two landing surfaces are associated with smaller probabilities than those associated with the three larger planar surfaces 602. The remnant taper 601 near these two additional surfaces is more highly curved than the main-body curved surfaces of the die, which, combined with the smaller areas of these two additional planar surfaces in comparison to the three larger planar surfaces 602, reduce the probability of the die landing on one of these two additional surfaces. In rolling experiments with prototype die, it was observed that die of the approximate shape illustrated in FIG. 6A land on one of the larger three surfaces 90% of the time with approximately equal probability. The die lands on one of the smaller two planar surfaces approximately 10% of the time.

FIG. 6B provides an end view of die 600 and the three larger planar surfaces 602 and 602′ designed to provide three different but equally probably outcomes. Surface 602′ is the landing surface, in this illustration. Numbers, dots, markings, or other indicia 603 are provided on the curved surface 604 opposite the three planar surfaces.

One of the smaller two surfaces 605, shown in FIG. 6B, includes an additional indicium or indicia which indicates an outcome when the die comes to rest on the opposite, complementary smaller surface.

FIG. 6C provides a perspective view of the die 600 in a landed position. FIG. 6D provides a view of the gaming die 600 having landed on one of its smaller end surfaces 605. The diameters of the ends are shown as both being equal and having the value D1. In contrast, FIG. 6E show a derivative die 607, also landed on one of the end surfaces 605“, in which the two end surfaces 605 and 605” have two different diameters D1 and D2, respectively. This results in die 607 having three equally probable primary landing positions and two different landing positions of unequal probabilities both smaller than those of the primary landing positions.

FIGS. 7A-E show a die with 4 primary and equally probable landing surfaces and 2 additional surfaces having at least one lower probability. FIG. 7A provides a top or side view of an ellipsoid-based gaming die 700 having four larger planar surfaces 702. FIG. 7A also shows that the ends of die 700 have been truncated to form smaller planar surfaces on the ends 704 that together comprise two additional landing positions. The remnant taper 701 on the ends is more highly curved than the main-body curved surfaces. This greater curvature, combined with the smaller areas of the two additional surfaces, combine to reduce the probability of the die landing on one of these two additional surfaces. Numbers, markings, or other Indicia 703 are provided on the planar surfaces 702.

In rolling experiments with prototype die, it was observed that die of the approximate shape illustrated in FIG. 7A landed on one of the four larger surfaces approximately 90% of the time with approximately equal probabilities. The die landed on one of the two smaller, additional surface approximately 10% of the time.

FIG. 7B provides an end view of die 700 and the four identically shaped and sized planar surfaces 702 designed to provide four different but equally possible outcomes. The die 700 in FIG. 7B is shown with an additional indicium or indicia on one of the smaller two surfaces.

FIG. 7C provides a perspective view of the die 700 in a high probability landed position and FIG. 7D provides a view of the gaming die 700 having landed in one of two lower but equal probability surfaces. The diameters of the end surfaces are shown as both being equal and both having the diameter D1. In contrast, FIG. 7E shows a derivative die 706 with end surfaces having two different diameters D1 and D2 end surfaces 704 and 704′. This results in die 707 having four equally probable primary landing positions and two different additional landing positions of lower and unequal probability.

FIGS. 8A-E shows a die with 5 primary and equally probable landing surfaces and 2 additional surfaces having at least one lower probability. FIG. 8A provides a top view of an ellipsoid-based gaming die 800 having five planar surfaces 803 (only three planar surfaces are evident in this view) with a number, marking, or other indicia 801 indicating a roll result placed or inscribed on curved surface 801. FIG. 8A also shows that the ends of die 800 have been truncated to generate end planar surfaces 805 that comprise two additional landing positions. The remnant taper 801 on the ends is more highly curved than that on the main-body surface of the die. The greater curvature and smaller surface area of the end surfaces combine to reduce the probability of the die landing on one of these two additional surfaces. In rolling experiments with prototype die, it was observed that die of the approximate shape illustrated in FIG. 8A land on one of the larger, five planar surfaces approximately 90% of the time with approximately equal probability. The die landed on one of the two end surfaces approximately 10% of the time.

FIG. 8B provides an end view of die 800 and the five identically sized and shaped planar surfaces 803 designed to provide five different but equally probable outcomes. Indicia 804 are provided on the curved surfaces 801. The die 800 is shown with ends truncated to generate two additional planar end surfaces 805 which may include additional indicia, such as indicium 806. FIG. 8C provides a perspective view of the die 800 in a higher probability landed position.

FIG. 8D provides a view of the gaming die 800 having landed on one of its end surfaces 805. The diameters of the ends are both D1. In contrast, FIG. 8E show a derivative die with end surfaces 805′ and 805″ having two different diameters D^(x) and D^(y), respectively

FIG. 8B provides an end view of die 800 and the five identically shaped and sized planar surfaces 803 designed to provide three different but equally possible outcomes. A number, marking, or other indicia on the curved side of the die 801 opposite the face down planar surface 802′ indicates the outcome in the illustrate orientation. The die 800 is shown with truncated ends that generate two additional planar end surfaces 805 associated with smaller probabilities. They may also include numbers, markings, or other indicia. FIG. 8C provides a perspective view of the die 800 in a higher probability landed position.

FIG. 8D provides a view of the gaming die 800 having landed on one of its end surfaces 805. The diameters of the ends are shown as both D1. In contrast, FIG. 6E shows a derivative die 806 similarly oriented but with end surfaces 605 and 605 having two different values D1 and D2. This results in die 806 having five equally probable primary landing positions and two different landing positions with lower and unequal probabilities.

FIGS. 9A-B shows a polyhedron die with 4 primary and equally probable landing surfaces and 4 additional surfaces having at least one lower probability. FIG. 9A provides a perspective view of a polyhedron, 900, more specifically a truncated tetrahedron, having eight faces that include four faces having equilateral-triangular shapes 902 a and 902 b (the other two faces are not shown in FIG. 9A) and four faces having equilateral-hexagonal shapes 903 a and 903 b. The die 900 has two types of stable landing positions, landing either on one of the hexagonal faces, as shown, or alternatively landing on one of the smaller triangular faces. In empirical studies of the die, it was determined that the die lands on one of the smaller triangular faces approximately 4% of the time and on one of the larger hexagonal faces approximately 96% of the time, two different types of outcomes with different probabilities for the same die.

FIG. 9B provides a perspective view of a polyhedral die. The polyhedral 901 die is again a truncated tetrahedron, having eight faces that include four faces having equilateral triangular shapes 904 a and 904 b and four faces having hexagonal shapes 905 a and 905 b. A difference between die 901 and die 900 is that one of the triangular faces, 904 a, is larger than the other three triangular faces. In addition, only one of the hexagonal faces is equilateral, shown in FIG. 9B as the landing surface. The other three hexagonal faces are irregular hexagons. As with die 900, die 901 has more than one probable stable landing positions, landing on the regular hexagonal base as shown or alternatively landing on any one of the other three irregular hexagonal faces. Die 901 can also land on any one of the equilateral triangular faces, however, a logical observer would conclude that the larger area triangle surface would be landed on with greatest frequency.

The used of truncated tetrahedron is instructional only. It will be obvious to the mindful observer than the modification of other polyhedrons can result in similar variations, including a cube where in reducing on of the 3 equal widths of the die, rendering it a rectangular solid, will change the odds of it landing on any surface.

While various embodiments have been described above, it should be understood that they have been presented by way of example only, and that the breadth and scope of the invention should not be limited by any of the above described exemplary embodiments. As one example, the sizes, dimensions, and angles between the surfaces and faces of the described die may be altered and modified to render different probabilities to the surfaces and shapes. Ellipsoids with different ratios between major and minor axes of symmetry may be employed. Any suitable material can be used to form the die. However, plastics which can be molded by injection, compression or casting are easiest for production. To design and form a gaming die based on an ellipsoid, the basic shape is provided with at least one planar surface (i.e., a circular solid segment of the ellipsoid removed) either on one of its tapered ends or along its length. The planar surface (nominally round in shape when viewed perpendicular to the surface) provides a stable landing position for the die in order for the user to determine more exactly the resulting object, mark or indicia at the end of a roll of die of the base shape.

The planar surfaces may be machined or otherwise formed following casting or molding of the die or may be molded or otherwise formed during manufacture. For example, the ellipsoid shape could be employed without having any planar surface but instead having a construction similar to that described earlier for the round gaming die, including a hollow core and ball bearing which assures the die lands in a definitive position. For example, if the desired response is 1, 2, 3, the interior can be shaped as a triangular pyramid assuring the ball bearing drops into one the vertices. If the desired response is 1,2,3,4 the interior can be shapes as a square based pyramid or, in the case of 1 through 5, the interior can be shaped as a hexagonal pyramid. This specification should be read and considered in accordance with the following claims and their equivalents. The specification and drawings are, accordingly, to be regarded in an illustrative for teaching the concept rather than a restrictive sense. 

1. A die comprising: a die body with a non-spherical, curved surface and two or more planar landing surfaces; and a result indication associated with each planar landing surface, each result indication prominently displayed when the die rests in a stable position on the planar landing surface with which the result indication is associated.
 2. The die of claim 1 wherein the non-spherical, curved surface of the die body comprises a portion of the surface of one of: an ovoid-shaped solid; a prolate-spheroid-shaped solid; and an ellipsoid-shaped solid.
 3. The die of claim 1 wherein the result indication associated with a planar landing surface is located at a position at the intersection of the surface of the die body and a line normal to the planar landing surface that passes through the center of the planar landing surface and continues through the die body to the position of the result indication.
 4. The die of claim 1 wherein each result indication comprises a numeral, marking, or indicia that is inscribed in, molded onto, printed on, painted on, and/or affixed to the surface of the die in a neighborhood surface about the result-indication position.
 5. The die of claim 1 wherein the landing surfaces are symmetrically disposed about a rotation symmetry axis of a solid, with which the curved surface of the die body is coincident.
 6. The die of claim 5 wherein the landing surfaces have a common shape and size and are associated with equal landing-surface probabilities.
 7. The die of claim 5 including a number landing surfaces equal to one of: 2; 3; 4; 5; and
 6. 8. The die of claim 1 wherein a first set of two or more of the landing surfaces are symmetrically disposed about a rotation symmetry axis of a solid, with which the curved surface of the die body is coincident; and wherein a second set of two of the landing surfaces are normal to the rotation symmetry axis of the a solid, with which the curved surface of the die body is coincident.
 9. The die of claim 8 wherein the landing surfaces include: a first set of two or more landing surfaces that have a common first shape and first size and are associated with equal first landing-surface probabilities; and a second set of two or more landing surfaces that have a common second shape and second size and are associated with equal second landing-surface probabilities that differ from the first landing-surface probabilities.
 10. The die of claim 8 wherein the second set of landing surfaces includes 2 landing surfaces; and wherein the first set of landing surfaces includes a number landing surfaces equal to one of: 2; 3; 4; 5; and
 6. 10. A die comprising: a polyhedral die body two or more planar landing faces of a first common size and shape and two or more additional planar landing surfaces of a second, common size and shape; and a result indication associated with each planar landing surface and planar landing face, each result indication prominently displayed when the die rests in a stable position on the planar landing surface with which the result indication is associated.
 11. The die of claim 10 wherein the two or more planar landing faces correspond to faces of one of a tetrahedron, cube, octahedron, dodecahedron, and icosahedron.
 12. The die of claim 1 wherein the result indication associated with a planar landing face is located on an opposite, complementary planar landing face.
 13. The die of claim 1 wherein the result indication associated with a planar landing surface is located on an opposite, complementary planar landing surface.
 14. A virtual die that is displayed on a display device, the virtual die comprising: a die body with a non-spherical, curved surface and two or more planar landing surfaces; and a result indication associated with each planar landing surface, each result indication prominently displayed when the die rests in a stable position on the planar landing surface with which the result indication is associated.
 15. The virtual die of claim 14 wherein the non-spherical, curved surface of the die body comprises a portion of the surface of one of: an ovoid-shaped solid; a prolate-spheroid-shaped solid; and an ellipsoid-shaped solid.
 16. The virtual die of claim 14 wherein the result indication associated with a planar landing surface is located at a position at the intersection of the surface of the die body and a line normal to the planar landing surface that passes through the center of the planar landing surface and continues through the die body to the position of the result indication.
 17. The virtual die of claim 14 wherein the landing surfaces are symmetrically disposed about a rotation symmetry axis of a solid, with which the curved surface of the die body is coincident.
 18. The virtual die of claim 17 wherein the landing surfaces have a common shape and size and are associated with equal landing-surface probabilities; and wherein the virtual die includes a number landing surfaces equal to one of: 2; 3; 4; 5; and
 6. 19. The virtual die of claim 14 wherein a first set of two or more of the landing surfaces are symmetrically disposed about a rotation symmetry axis of a solid, with which the curved surface of the die body is coincident; and wherein a second set of two of the landing surfaces are normal to the rotation symmetry axis of the a solid, with which the curved surface of the die body is coincident.
 20. The virtual die of claim 8 wherein the landing surfaces include a first set of two or more landing surfaces that have a common first shape and first size and are associated with equal first landing-surface probabilities; and a second set of two or more landing surfaces that have a common second shape and second size and are associated with equal second landing-surface probabilities that differ from the first landing-surface probabilities; wherein the second set of landing surfaces includes 2 landing surfaces; and wherein the first set of landing surfaces includes a number landing surfaces equal to one of: 2; 3; 4; 5; and
 6. 